In differential geometry, a metaplectic structure is the symplectic analog of spin structure on orientable Riemannian manifolds.
A metaplectic structure on a symplectic manifold allows one to define the symplectic spinor bundle, which is the Hilbert space bundle associated to the metaplectic structure via the metaplectic representation, giving rise to the notion of a symplectic spinor field in differential geometry.
Symplectic spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in establishing the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology.
They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory.
They form the foundation for symplectic spin geometry.
A metaplectic structure [1] on a symplectic manifold
is an equivariant lift of the symplectic frame bundle
with respect to the double covering
{\displaystyle \rho \colon {\mathrm {Mp} }(n,{\mathbb {R} })\to {\mathrm {Sp} }(n,{\mathbb {R} }).\,}
In other words, a pair
is a metaplectic structure on the principal bundle
when The principal bundle
is also called the bundle of metaplectic frames over
Two metaplectic structures
on the same symplectic manifold
, ω )
are called equivalent if there exists a
are two equivalent double coverings of the symplectic frame
of the given symplectic manifold
Since every symplectic manifold
is necessarily of even dimension and orientable, one can prove that the topological obstruction to the existence of metaplectic structures is precisely the same as in Riemannian spin geometry.
[2] In other words, a symplectic manifold
admits a metaplectic structures if and only if the second Stiefel-Whitney class
In fact, the modulo
reduction of the first Chern class
is the second Stiefel-Whitney class
admits metaplectic structures if and only if
If this is the case, the isomorphy classes of metaplectic structures on
are classified by the first cohomology group
is assumed to be oriented, the first Stiefel-Whitney class